Foucart, Clément, Ma, Chunhua and Yuan, Linglong 
ORCID: 0000-0002-7851-1631
(2022)
Limit theorems for continuous-state branching processes with immigration.
Advances in Applied Probability, 54 (2).
pp. 599-624.
ISSN 0001-8678, 1475-6064
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Abstract
<jats:title>Abstract</jats:title><jats:p>A continuous-state branching process with immigration having branching mechanism<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0001867821000434_inline1.png"/><jats:tex-math>$\Psi$</jats:tex-math></jats:alternatives></jats:inline-formula>and immigration mechanism<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0001867821000434_inline2.png"/><jats:tex-math>$\Phi$</jats:tex-math></jats:alternatives></jats:inline-formula>, a CBI<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0001867821000434_inline3.png"/><jats:tex-math>$(\Psi,\Phi)$</jats:tex-math></jats:alternatives></jats:inline-formula>process for short, may have either of two different asymptotic regimes, depending on whether<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0001867821000434_inline4.png"/><jats:tex-math>$\int_{0}\frac{\Phi(u)}{|\Psi(u)|}\textrm{d} u<\infty$</jats:tex-math></jats:alternatives></jats:inline-formula>or<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0001867821000434_inline5.png"/><jats:tex-math>$\int_{0}\frac{\Phi(u)}{|\Psi(u)|}\textrm{d} u=\infty$</jats:tex-math></jats:alternatives></jats:inline-formula>. When<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0001867821000434_inline6.png"/><jats:tex-math>$\int_{0}\frac{\Phi(u)}{|\Psi(u)|}\textrm{d} u<\infty$</jats:tex-math></jats:alternatives></jats:inline-formula>, the CBI process has either a limit distribution or a growth rate dictated by the branching dynamics. When<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0001867821000434_inline7.png"/><jats:tex-math>$\scriptstyle\int_{0}\tfrac{\Phi(u)}{|\Psi(u)|}\textrm{d} u=\infty$</jats:tex-math></jats:alternatives></jats:inline-formula>, immigration overwhelms branching dynamics. Asymptotics in the latter case are studied via a nonlinear time-dependent renormalization in law. Three regimes of weak convergence are exhibited. Processes with critical branching mechanisms subject to a regular variation assumption are studied. This article proves and extends results stated by M. Pinsky in ‘Limit theorems for continuous state branching processes with immigration’ (<jats:italic>Bull. Amer. Math. Soc.</jats:italic><jats:bold>78</jats:bold>, 1972).</jats:p>
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | 4901 Applied Mathematics, 49 Mathematical Sciences, 4905 Statistics |
| Divisions: | Faculty of Science and Engineering > School of Physical Sciences |
| Depositing User: | Symplectic Admin |
| Date Deposited: | 07 Jul 2022 15:26 |
| Last Modified: | 06 Dec 2024 19:51 |
| DOI: | 10.1017/apr.2021.43 |
| Related URLs: | |
| URI: | https://livrepository.liverpool.ac.uk/id/eprint/3157886 |
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